On q-analogues of partially ordered sets

Abstract

We define a notation of an f-q-analogue of a poset P, were f is a function in the incidence algebra of P. In this setting, for given f and q, all f-q-analogues have the same Zeta and characteristic polynomials, and the same Möbius invarints for rank selections. These are q-analogues of the corresponding entities for th eposet P. We describe conditions when P is being shellable implies that its f-q-analogues are also shellable. In such situations, the analogues admit a shelling pulled back in a natural way from one of P, revealing a natural projection from the homology of analogues to that of P. As a by-product we obtain the non-negativity of the coefficients of the Betti polynomial for the analogues and their rank-selected subposets. We discuss the behavior of f-q-analogues with respect to several operations on the function f, the value q, and the poset P. Examples include posets of set partitions, posets of shuffles, semimodular and distributive lattices, and products of chains in particular. This work is an attempt to unify recent approaches to order analogues, and integrates cases studied by Butler; Björner and Stanley; and Bennett, Dempsey and Sagan.